# Properties

 Label 2.61.az_kl Base Field $\F_{61}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{61}$ Dimension: $2$ L-polynomial: $1 - 25 x + 271 x^{2} - 1525 x^{3} + 3721 x^{4}$ Frobenius angles: $\pm0.0746782053317$, $\pm0.283932848163$ Angle rank: $2$ (numerical) Number field: 4.0.1639109.1 Galois group: $D_{4}$ Jacobians: 5

This isogeny class is simple and geometrically simple.

## Newton polygon

This isogeny class is ordinary.

 $p$-rank: $2$ Slopes: $[0, 0, 1, 1]$

## Point counts

This isogeny class contains the Jacobians of 5 curves, and hence is principally polarizable:

• $y^2=31x^6+14x^5+19x^4+53x^3+50x^2+48x+39$
• $y^2=23x^6+4x^5+15x^4+44x^3+46x^2+35x+6$
• $y^2=47x^6+42x^5+46x^4+46x^3+59x^2+36x+23$
• $y^2=29x^6+55x^5+50x^4+44x^3+31x^2+55x+33$
• $y^2=6x^6+5x^5+18x^4+35x^3+6x^2+19x+35$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2443 13541549 51548582575 191740222452149 713336156254883728 2654331215399083968125 9876822194764167581661583 36751691963860975798195584869 136753054580313921280528613143675 508858111517766385885362591180873984

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 37 3639 227107 13848219 844588302 51520029663 3142739546347 191707303124019 11694146241606517 713342914323716854

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{61}$
 The endomorphism algebra of this simple isogeny class is 4.0.1639109.1.
All geometric endomorphisms are defined over $\F_{61}$.

## Base change

This is a primitive isogeny class.

## Twists

Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.61.z_kl $2$ (not in LMFDB)